Is it possible to rig an STL file?
No, it is not possible to rig an STL file.

Firstly, Blender cannot open STL files. They can only be imported to Blender and once imported the information contained in such files is translated to a format native to Blender and, so to speak, become a .blend file. 

Secondly, the term 'rigging' in CG context does not apply to files or file formats and saying to rig a file format makes no sense at all. Rigging is a process of making some kind of deformation or control system to make it easier to animate objects in animation software packages like Blender. So if you want to rig a 3d model that you imported from an  STL file to Blender, then it's exactly the same as to rig any other model in Blender. Rigging however is a very complex and wide subject and cannot be described in full in one answer. You will need to ask a more specific question and provide the full context of your situation to get any meaningful help.